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Full text of "Treatise On Analysis Vol-Ii"

436      XV    NORMED ALGEBRAS AND SPECTRAL THEORY

for n 2> 0, which are all strictly positive, show that P0() == 1 and that

(3)                         - '^              l

C2        *      Cn + 1
Cn       *      C2n-i

1

for all n > 1. Also we have

(4)

(e) Conversely, for every (bounded or unbounded) positive measure ^ on R, show
that the operator MM (which is in general unbounded) is self-adjoint and simple.

2. Conversely, let / be a Jacobi matrix (formula (1) of Problem 1) in which the an are
real and the bn are real and >0, but are not restricted in any other way. Let (en)n^o
be the canonical Hilbert basis of the Hilbert space / (6.5). Define a hermitian
operator H, with domain the subspace G of / generated by the (finite) linear com-
binations of the en, by the conditions

H- en = bn-i en-i + anen + bnen^       (n ^0)

(with e~i = 0, _! = 0). By abuse of notation, let H also denote the closure of the
hermitian operator H.

00

(a)    Let  be a complex number. If y =  yn en is an eigenvector of H* corresponding

to the eigenvalue , show that yn = Pn(QyG, where Pn is a polynomial of degree /z,
determined recursively by the formulas PO() = 1 and

for all n 2> 0. Deduce that the defect of His (1, 1) if, for some  e C such that ./ ^ 0,
we have

00

~~       "    ; < + oo:

in which case this inequality is true for all  e C such that ./ = 0. In the contrary
case, the defect of H is (0,0), in other words H is self-adjoint (having regard to the
abuse of language mentioned above). In every case there exist self-adjoint extensions
ofH.

(b)    Let F be the subspace of ^C(R) consisting of polynomials with complex coeffi-
cients. Show that there exists a unique linear form ap on F such that

ap(Pn Pm) = $mn      (Kronecker index).

(If a polynomial is of the form RS, where R = ]T ut Pt and S = ]T vt PI , then we must
have aF(RS) =2 * ty, which shows that aF is unique. For the existence, it is enough

to show that, if we write the product U(jc) = (x  z0)R(jc)S(x) in the two forms
((*  z0)R(x))S(x) and R(#)((#  z0)S(x)), where z0 is any complex number, then the
calculation of aF(U) from each of these two products must give the same value.
Reduce to the case where each of R, S is one of the polynomials Pn, and use the
relation (1).) (15.12.8),